How can I solve this PDE $h(\phi_{xx}+\phi_{yy})+h_{x}\phi_{x}+h_{y}\phi_{y}=0, h=5-e^{-x^4-y^2}$

Question

$\phi(x,y)$ is the velocity potential ($\frac{\partial \phi}{\partial x}=-u,\frac{\partial \phi}{\partial y}=-v$) for a steady flow in a 2D ocean. $h(x,y)=5-e^{-x^4-y^2}$ is the water depth at $(x,y)$. Mass conservation leads to the following PDE for $\phi$: $$h(\frac{\partial ^{2}\phi}{\partial x^{2}}+\frac{\partial ^{2}\phi}{\partial y^{2}})+\frac{\partial h}{\partial x}\frac{\partial \phi }{\partial x}+\frac{\partial h}{\partial y}\frac{\partial \phi }{\partial y}=0$$ Solution domain is: $$-5\leq x \leq 5, \ -5\leq y \leq 5 $$ At all four boundaries flow is horizontal i.e. $u=u_{0},v=0$. So, boundary condotions will be: $$ \phi(-5,y)=-u_{0}x,\ \ \ \phi(+5,y)=-u_{0}x$$ $$ \phi(x,-5)=-u_{0}x \ \ \ \phi(x,+5)=-u_{0}x$$ I couldn't solve this equation using separable variables. Is there any other way to solve this equation? any analytical method such as Fourier series, power series, integral transforms, etc. is accepted. I would really appreciate it if anyone could help me.